ON A FAMILY OF SURFACES OF REVOLUTION OF ... - Project Euclid

J. ARROYO, O. J. GARAY AND J. J. MENCIA KODAI MATH. J. 21 (1998), 73-80

ON A FAMILY OF SURFACES OF REVOLUTION OF FINITE CHEN-TYPE* To the memory of Prof. I. Rozas J. ARROYO, O. J. GARAY AND J. J. MENCIA

Abstract We study Chen-finite type surfaces of revolution in E3 which contain two affine circles through each point. First, we prove that they can be obtained by revolving an ellipse on a suitable axis and then we show that only the 2-dimensional sphere is of finite type.

1.

Introduction

Euclidean submanifolds of finite type were introduced by B. Y. Chen in the late seventies and it has been a topic of active research since then (for details see [3] [4]). An Euclidean submanifold is said to be of Chen finite type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian, [3]. Compact Euclidean submanifolds are characterized by both a polynomial criterium and also by satisfying a variational principle, [4]. B. Y. Chen posed in [5] [6] the problem of classifying the finite type surfaces in the 3-dimensional Euclidean space E3. In fact the only known finite Chen-type surfaces in E3 are portions of spheres, circular cylinders and of minimal surfaces and it was B. Y. Chen who made in [6] the following conjecture. CONJECTURE.

The only compact surfaces of finite type in E3 are the spheres.

The conjecture is still unsolved but it has been confirmed by different authors by proving that finite type tubes, finite type ruled surfaces, finite type quadrics, finite type cones and finite type cyclides of Dupin are surfaces of the only known examples in E3, [4]. However, for another classical family of surfaces in E3, the surfaces of revolution, the classification of its finite type members is not known yet. Particular cases of this problem were consider in [7], [8] and [9]. In [7] two

* Partially supported by a grant of Gobierno Vasco PI 95/95 Received November 11, 1997; revised February 4, 1998. 73

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J. ARROYO, O. J. GARAY AND J. J. MENCIA

families of finite type surfaces of revolution with generating curve satisfying certain algebraic conditions are classified. In [8] the conjecture was confirmed for surfaces of revolution with finite type coordinate functions. Finally in [9], authors gave the classification of surfaces of revolution with constant mean curvature. Our purpose here is to classify a family g of compact finite type surfaces of revolution M2 which is determined by satisfying the following geometric property: (P) There are at least two affine circles {ellipses) contained in M2 e g through each point p e M2. As a motivation for the study of the above property we remind that in 1822 C. Dupin defined a cyclide to be a surface M2 in E3 which is the envelope of a family of spheres tangent to three fixed spheres. All compact cyclides of Dupin in E3 can be obtained by inversion of a torus of revolution and, therefore, they contain four metric circles through each point [2], In 1980 R. Blum gave an example of compact cyclides with 4, 5 and 6 metric circles through each point [1] and K. Ogiue and N. Takeuchi [10] proved that a compact surface of revolution which contains at least two metric circles through each point, is a Hulahoop surface. Hulahoop surfaces have 4, 5 or infinitively many metric circles through each point. In order to achieve our goal, we begin with defining the elliptical hulahoop surfaces as those surfaces of E3 obtained by revolving an ellipse around a suitable axis. We first prove 1. Let M2 be a compact surface of revolution in E3 which contains at least two ellipses through each point, then it is an elliptic hulahoop surface. PROPOSITION

This result is analogous to the one proved in [10] for metric circles, but their method can not be applied here so that we need to furnish a different argument. Note also that our family includes the hulahoop surfaces of [10]. Then we will prove PROPOSITION

2. The only elliptic hulahoop surface of finite type is the sphere.

As a consequence COROLLARY 1. The only compact finite type surface of revolution which satisfies (P) is the sphere.

This confirms Chen's conjecture.

SURFACES OF REVOLUTION

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2. Preliminaries Let x: Mn —> Em be an isometric immersion of a compact Riemannian manifold into the w-dimensional Euclidean space. The inmersion (M ,JC) is said to be of finite Chen-type k if the position vector x admits the following spectral decomposition Λ

k

(1)

x = xo

where xt are 2sw-valued eigenfunctions of the Laplacian of {Mn,x): Axt = λtxt. The following minimal polynomial criterion is a useful tool to decide whether a submanifold is of finite Chen-type [3]. 3. An isometric inmersion (Mn,x) of a compact Riemannian manifold in E is of finite Chen-type, if and only if there exists a nontrivial polynomial Q(t) such that Q(A)(x — xo) = 0, where Δ is the Laplacian of (Mn,x). PROPOSITION m

Now, we want to define the elliptic hulahoop surfaces of E3. These are surfaces of revolution which are obtained by revolving an ellipse around an axis which is not perpendicular to the plane containing the ellipse. They can be obtained in the following way: Choose the x,y,z axes so that the first two are parallel to the axes of the ellipse E{a,b,r,s)r, s>0 given by 2

(x-a) r2

2

+

(y-b) s2

which is supposed to be contained in the cy-plane. Rotate the ellipse around its centre by an angle β e [0,π/2] and denote by E(a,b,r,s,β) the resulting ellipse. Finally we denote by E(a,bJris,β1(x) the ellipse obtained by tilting E(a,b,r,s,β) around a diameter parallel to the x-axis by an angle α e [0,π]. It can be easily checked that the final ellipse is parametrized by

{

x = a-\-rcosβcosθ — .ssin/?sin0, y~b + cosα(rsin/?cos# 4- scosβάnθ), z = sin α(r sin β cos θ + s cos β sin θ),

with 0e[O,2π]. Now let H(a,b,r,s,β,cc) be the surface which is obtained by revolving E(a^b1r1s1β,oί) around the z-axis. Then one can check by lengthly computation that H is a regular surface, if and only if, either (i) 0 Φ a Φ nil, 0 / b2d2 + a2 cos2 cue2 + labe cos α - r2s2 cos2 α, 0 > e2 - a2c2 or

76


(ii) α = π/2, b Φ 0, 0 > e2 - a2c2 or (iii) α = π/2, b = 0, rf2 < 0 2 or (iv) α = π/2, a = b = 0, r = s or (v) α = π/2, a = b = 0, β = 0 where rf2 = r 2 cos 2 β + s2 sin2 β c 2 = s2 cos2jff + r2 sin 2 βsinde= (s2 - r2) sin jff cos £. 3. Proofs of the results First we give: Proof of Proposition 1. Since it is a compact surface of revolution, M2 must be either a topological sphere or a topological torus. First, we consider the simply connected case. Let po be any point of M2. Since there are two ellipses through po, at least one of them, Eo, is not a latitude. If the latitude Co through po is not transversal to Eo, we can take a point on Eo, p\, as close to po as necessary, so that the latitude C\ through p\ cuts Eo transversally. Take p2 = Eo n C\ - {p\}. Let Π be the perpendicular bisector of the segment pϊpi. Since pipϊ forms a chord of C\, Π must contain the axis of revolution. Thus we can consider γ = Tln M2 as the plane generating curve of M2. Let us denote by N the part of M2 which is generated by the points of Eo. Clearly Π n N is a closed segment β of γ. By regularity, one can see that β must be symmetric with respect to Π and then it can be checked by direct computation that β must be a piece of a conic. Now by using a similar argument for the border points of β and using regularity and simply connectedness of M2, we can 2 conclude that M must be a ellipsoid of revolution. 2 Now, let us assume that M is of genus one. Let 5£ and M be homotopy classes corresponding to a latitude and a meridian respectively. Then J&? and J( generate the fundamental group π\{M2) of M2. It is known [11] that an ellipse contained in M2 must be in one of the following homotopy classes Θ9 «£?, M9 S£ + Jt or 5£ - Jί, where 0 is the trivial homotopy class. We wish to prove that M2 satisfies the following property: (*) There exists α point p e M2 such that one of the ellipses through p which is not a latitude, does not belong to ΘKJ £?. 2

In fact, take po e M and suppose it is not a point satisfying (*), and consider an ellipse Eo through po which belongs to 0 u S£. Then through almost every point


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of iso there passes a latitude Co which cuts Eo in two different points. By using an argument analogous to that of the simply connected case, we can see that there is a closed piece of a conic β which is included in the copy of the generating curve of M2, γ, which passes through /?o Take now a point p\ of the border of β. If M2 satisfies (•*) at p\, we have done. If not, we obtain for p\ the same conclusion as for po and, therefore, by regularity, there exists a closed piece of a conic, which we also denote by β, that passes through po and p\ and is contained in γ. By repeating the process we see that either exists a point where M2 satisfies (*) or γ must be an ellipse through po, but the latter contradicts the election of po This shows (*). Then there exists an ellipse E which belongs to one of Jί, S£ + Jt or S£ — M. This means that M2 is swept by E under rotation. Q.E.D. Now, suppose that we have an elliptic hulahoop surface H(a,b,r,s,β,oc) (satisfying some of the regularity conditions (i) to (iv)). If we put (3)

q{θ) = (a + r cos)S cos (9 - . + (b + r sin/? cos α cos θ + s cos β cos α sin θ)2

then we observe that H(a,b,r,s,β1oc) is obtained by revolving the curve of the xz-plane given by (4)

ix={q{θ))X'2 | j = 0,

y z= with 0e[O,2π). For a surface of revolution parametrized by (5)

x = (u(θ) cos φ, u(θ) sin φ, v(θ))

one can check by a straight-forward computation that the Laplace's operator is given by 2

[ )

1 d 2 hdθ

IV \uh

u'u" + υ'v"\ d 2 h jdθ

2

1 d 2 2 u dφ

where h = (u')2 + (t/) 2 . By using (6) for the parametrization given in (4), one can see that the Laplacian of H(a1b1 r,s,β, α) for an / e C°°(fΓ(έϊ,fe,r,j,jS, α)) which depends only on θ satisfies

where # denote the discriminant of the first fundamental form.

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Proof of Proposition 2. We do our discusion in terms of the regularity conditions (i) to (iv). We first consider the case r Φ s, then g(θ) can be expressed as (8)

2

2

2

g(θ) = I (r - s )(cos

2

2

2

2

2

2

2

β(s cos α - r ) 4- sin β(s - r cos α)) cos 40

o 2

2

2

2

2

2

2

+ - (r — s )rs sin/? cos/? sin α sin 40 - - (r - s )r{a cos/? 4- 6 sin/?cos α) cos 30 - - (r - s )s(-a

sin/? 4- Z> cos/? cos α) sin 30

Considering g and # as polynomials in cos0, sin0, we observe that 1 < deg(#) < 2 and that deg(#) = 1, if and only if, β = 0 and r = scosα, where deg(#) means degree of q. Also we have deg(#) = deg(#) 4- 2. One can obtain by induction the following formula for the higher order Laplacian of the function z given in (4) [?)

Δ

k

_ Akcosmθ +

-^j

z -

Bksmmθ+pk

K

> υ

where Ak,BkeR do not vanish simultaneously and /?jt(sin0,cos0) is a polynomial of degree at most m{k) — 1 with m(k)

= (3 deg(2r2 + 2r 4 sin 2 α

then deg(#) = 2 and 1 < deg(gf) < 2 with deg(#) = 1, if and only if, b = 0 and r 2 sin 2 α = a2 cos 2 α.


79

Again from (4), (7) and by induction one can check after a long computation

where ( )' is meant for the derivative with respect to θ, ak = (-23r2asmoc)(-l)k-1

1 4 7 ... (1 + 6(k - 1))

and p/c is a polynomial in sin0, cos0 of degree 2k + 1 + (2fc — 2) deg(#), A: > 1. If H(a,b,r,s,β,